One approach to local least-squares polynomial approximation of data in two independent variables consists of centring a square subarea on each point in a data set and computing the coefficients that define the best-fit polynomial over the subset defined by this window. Replacing the input sample at the centre of the window with the value of the best-fit polynomial enables formulation of an expression for the impulse responses of low-pass filters. This is simplified if the approximating function is written in terms of orthogonal polynomials. Cutoff frequencies of these filters are inversely proportional to the polynomial order. Polynomial order has been defined as the highest power either of any term in the polynomial or of any independent variable in the polynomial. The latter definition results in polynomials that consist of all of the terms that comprise polynomials arising from the former definition, as well as higher-order cross terms. Including these extra terms lessens the directional bias of the transfer functions. However, some distortion persists, with lowest frequencies passed in the direction of the coordinate axes and highest frequencies passed at 45 degrees to the axes. The dependency of cutoff frequency on polynomial order and window size has been empirically quantified. These results, coupled with the expressions for the impulse responses, make possible the design of convolution operators without transformation to the frequency domain. Application of these local least-squares operators to a Bouguer gravity data set from northern Alberta indicates that these produce regional anomaly fields that are generally similar to those obtained from wavelength filtering.
Approximating an entire data with one polynomial is commonly done to compute long-wavelength constituents of the Bouguer gravity field. When this technique is formulated in terms of convolution, a unique weight function operates on each data point. The operators for the central portion of the data set have amplitude responses that are similar to those of local least-squares operators. This implies that polynomial approximation of the long-wavelength components of the gravity field should incorporate the higher-order cross terms in order to diminish directional bias. This is demonstrated by applying this method to the aforementioned Bouguer gravity data.
The local least-squares filters have been transformed to operators that output the components of the horizontal-gradient vector. This combines the high-frequency amplification property of differentiation with the low-pass nature of local least-squares approximation. This generates operators with both variable passbands and a likeness to the ideal response in this passband. These gradient-component operators have been used to compile a suite of maps of the magnitude of the horizontal-gradient vector of the Bouguer gravity field, featuring different bandwidths of crustal anomalies. Because the procedure is formulated in terms of convolution, the frequency content of each magnitude map is known. This facilitates the generation of complementary pairs of gradient-magnitude and wavelength-filtered maps. The gradient-magnitude maps enhance the interpretation by highlighting density boundaries. This is useful for mapping the edges of source bodies, identifying subtle characteristics, and determining spatial relationships between trends.
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